Analysis I | Part IA, 2008

Let f:[0,1]Rf:[0,1] \rightarrow \mathbb{R} be Riemann integrable, and for 0x10 \leqslant x \leqslant 1 set F(x)=0xf(t)dtF(x)=\int_{0}^{x} f(t) d t.

Assuming that ff is continuous, prove that for every 0<x<10<x<1 the function FF is differentiable at xx, with F(x)=f(x)F^{\prime}(x)=f(x).

If we do not assume that ff is continuous, must it still be true that FF is differentiable at every 0<x<10<x<1 ? Justify your answer.

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